Langevin Dynamics (LD) and its discrete proposal have been widely applied in the field of Combinatorial Optimization (CO). Both sampling-based and data-driven approaches have benefited significantly from these methods. However, LD's reliance on Gaussian noise limits its ability to escape narrow local optima, requires costly parallel chains, and performs poorly in rugged landscapes or with non-strict constraints. These challenges have impeded the development of more advanced approaches. To address these issues, we introduce Fractional Langevin Dynamics (FLD) for CO, replacing Gaussian noise with $\alpha$-stable L\'evy noise. FLD can escape from local optima more readily via L\'evy flights, and in multiple-peak CO problems with high potential barriers it exhibits a polynomial escape time that outperforms the exponential escape time of LD. Moreover, FLD coincides with LD when $\alpha = 2$, and by tuning $\alpha$ it can be adapted to a wider range of complex scenarios in the CO fields. We provide theoretical proof that our method offers enhanced exploration capabilities and improved convergence. Experimental results on the Maximum Independent Set, Maximum Clique, and Maximum Cut problems demonstrate that incorporating FLD advances both sampling-based and data-driven approaches, achieving state-of-the-art (SOTA) performance in most of the experiments.

Branch-and-bound (B\&B) has long been favored for tackling complex Mixed Integer Programming (MIP) problems, where the choice of branching strategy plays a pivotal role. Recently, Imitation Learning (IL)-based policies have emerged as potent alternatives to traditional rule-based approaches. However, it is nontrivial to acquire high-quality training samples, and IL often converges to suboptimal variable choices for branching, restricting the overall performance. In response to these challenges, we propose a novel hybrid online and offline reinforcement learning (RL) approach to enhance the branching policy by cost-effective training sample augmentation. In the online phase, we train an online RL agent to dynamically decide the sample generation processes, drawing from either the learning-based policy or the expert policy. The objective is to strike a balance between exploration and exploitation of the sample generation process. In the offline phase, a value function is trained to fit each decision's cumulative reward and filter the samples with high cumulative returns. This dual-purpose function not only reduces training complexity but also enhances the quality of the samples. To assess the efficacy of our data augmentation mechanism, we conduct comprehensive evaluations across a range of MIP problems. The results consistently show that it excels in making superior branching decisions compared to state-of-the-art learning-based models and the open-source solver SCIP. Notably, it even often outperforms Gurobi.

Data plays a pivotal role in the development of both classic and learning-based methods for Mixed-Integer Linear Programming (MILP). However, the scarcity of data in real-world applications underscores the necessity for MILP instance generation methods. Currently, these methods primarily rely on iterating random single-constraint modifications, disregarding the underlying problem structure with constraint interrelations, thereby leading to compromised quality and solvability. In this paper, we propose ACM-MILP, a framework for MILP instance generation, to achieve adaptive constraint modification and constraint interrelation modeling. It employs an adaptive constraint selection mechanism based on probability estimation within the latent space to preserve instance characteristics. Meanwhile, it detects and groups strongly related constraints through community detection, enabling collective modifications that account for constraint dependencies. Experimental results show significant improvements in problem-solving hardness similarity under our framework. Additionally, in the downstream task, we showcase the efficacy of our generated instances for hyperparameter tuning. Source code is available: https://github.com/Thinklab-SJTU/ACM-MILP.

Graph matching aims to find node-to-node matching among multiple graphs, which is a fundamental yet challenging problem. To facilitate graph matching in scientific research and industrial applications, pygmtools is released, which is a Python graph matching toolkit that implements a comprehensive collection of two-graph matching and multi-graph matching solvers, covering both learning-free solvers as well as learning-based neural graph matching solvers. Our implementation supports numerical backends including Numpy, PyTorch, Jittor, Paddle, runs on Windows, MacOS and Linux, and is friendly to install and configure. Comprehensive documentations covering beginner's guide, API reference and examples are available online. pygmtools is open-sourced under Mulan PSL v2 license.

Graph matching (GM) aims at discovering node matching between graphs, by maximizing the node-and edgewise affinities between the matched elements. As an NP-hard problem, its challenge is further pronounced in the existence of outlier nodes in both graphs which is ubiquitous in practice, especially for vision problems. However, popular affinity-maximization-based paradigms often lack a principled scheme to suppress the false matching and resort to handcrafted thresholding to dismiss the outliers. This limitation is also inherited by the neural GM solvers though they have shown superior performance in the ideal no-outlier setting. In this paper, we propose to formulate the partial GM problem as the top-k selection task with a given/estimated number of inliers k. Specifically, we devise a differentiable top-k module that enables effective gradient descent over the optimal-transport layer, which can be readily plugged into SOTA deep GM pipelines including the quadratic matching network NGMv2 as well as the linear matching network GCAN. Meanwhile, the attention-fused aggregation layers are developed to estimate k to enable automatic outlier-robust matching in the wild. Last but not least, we remake and release a new benchmark called IMC-PT-SparseGM, originating from the IMC-PT stereomatching dataset. The new benchmark involves more scale-varying graphs and partial matching instances from the real world. Experiments show that our methods outperform other partial matching schemes on popular benchmarks.

Encoding constraints into neural networks is attractive. This paper studies how to introduce the popular positive linear satisfiability to neural networks. We propose the first differentiable satisfiability layer based on an extension of the classic Sinkhorn algorithm for jointly encoding multiple sets of marginal distributions. We further theoretically characterize the convergence property of the Sinkhorn algorithm for multiple marginals, and the underlying formulation is also derived. In contrast to the sequential decision e.g. reinforcement learning-based solvers, we showcase our technique in solving constrained (specifically satisfiability) problems by one-shot neural networks, including i) a neural routing solver learned without supervision of optimal solutions; ii) a partial graph matching network handling graphs with unmatchable outliers on both sides; iii) a predictive network for financial portfolios with continuous constraints. To our knowledge, there exists no one-shot neural solver for these scenarios when they are formulated as satisfiability problems. Source code is available at https://github.com/Thinklab-SJTU/LinSATNet.